author  paulson 
Mon, 19 Sep 2005 15:12:13 +0200  
changeset 17484  f6a225f97f0a 
parent 17412  e26cb20ef0cc 
child 17819  1241e5d31d5b 
permissions  rwrr 
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(* Author: Jia Meng, Cambridge University Computer Laboratory 
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ID: $Id$ 

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Copyright 2004 University of Cambridge 

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Transformation of axiom rules (elim/intro/etc) into CNF forms. 

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*) 

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signature RES_AXIOMS = 
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sig 

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exception ELIMR2FOL of string 

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val tagging_enabled : bool 
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val elimRule_tac : thm > Tactical.tactic 
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val elimR2Fol : thm > term 
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val transform_elim : thm > thm 
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val clausify_axiom_pairs : (string*thm) > (ResClause.clause*thm) list 
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val cnf_axiom : (string * thm) > thm list 
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val meta_cnf_axiom : thm > thm list 

16012  18 
val rm_Eps : (term * term) list > thm list > term list 
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val claset_rules_of_thy : theory > (string * thm) list 
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val simpset_rules_of_thy : theory > (string * thm) list 

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val claset_rules_of_ctxt: Proof.context > (string * thm) list 
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val simpset_rules_of_ctxt : Proof.context > (string * thm) list 
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val clausify_rules_pairs : 
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(string*thm) list > thm list > (ResClause.clause*thm) list list * thm list 
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val clause_setup : (theory > theory) list 
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val meson_method_setup : (theory > theory) list 

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end; 
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structure ResAxioms : RES_AXIOMS = 
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struct 

15347  32 

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val tagging_enabled = false (*compile_time option*) 
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(**** Transformation of Elimination Rules into FirstOrder Formulas****) 
15347  36 

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(* a tactic used to prove an elimrule. *) 
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fun elimRule_tac th = 
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((rtac impI 1) ORELSE (rtac notI 1)) THEN (etac th 1) THEN 

16588  40 
REPEAT(fast_tac HOL_cs 1); 
15347  41 

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exception ELIMR2FOL of string; 

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(* functions used to construct a formula *) 
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fun make_disjs [x] = x 
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 make_disjs (x :: xs) = HOLogic.mk_disj(x, make_disjs xs) 
15347  48 

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fun make_conjs [x] = x 

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 make_conjs (x :: xs) = HOLogic.mk_conj(x, make_conjs xs) 
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fun add_EX tm [] = tm 

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 add_EX tm ((x,xtp)::xs) = add_EX (HOLogic.exists_const xtp $ Abs(x,xtp,tm)) xs; 

15347  54 

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fun is_neg (Const("Trueprop",_) $ (Const("Not",_) $ Free(p,_))) (Const("Trueprop",_) $ Free(q,_)) = (p = q) 
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 is_neg _ _ = false; 
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15347  60 

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exception STRIP_CONCL; 

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fun strip_concl' prems bvs (Const ("==>",_) $ P $ Q) = 
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let val P' = HOLogic.dest_Trueprop P 
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val prems' = P'::prems 

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in 

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strip_concl' prems' bvs Q 
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end 
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 strip_concl' prems bvs P = 
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let val P' = HOLogic.Not $ (HOLogic.dest_Trueprop P) 
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in 

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add_EX (make_conjs (P'::prems)) bvs 
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end; 
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fun strip_concl prems bvs concl (Const ("all", _) $ Abs (x,xtp,body)) = strip_concl prems ((x,xtp)::bvs) concl body 

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 strip_concl prems bvs concl (Const ("==>",_) $ P $ Q) = 

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if (is_neg P concl) then (strip_concl' prems bvs Q) 

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else 

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(let val P' = HOLogic.dest_Trueprop P 
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val prems' = P'::prems 
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in 

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strip_concl prems' bvs concl Q 

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end) 

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 strip_concl prems bvs concl _ = add_EX (make_conjs prems) bvs; 

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fun trans_elim (main,others,concl) = 
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let val others' = map (strip_concl [] [] concl) others 

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val disjs = make_disjs others' 
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in 

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HOLogic.mk_imp (HOLogic.dest_Trueprop main, disjs) 
15347  95 
end; 
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(* aux function of elim2Fol, take away predicate variable. *) 
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fun elimR2Fol_aux prems concl = 
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let val nprems = length prems 
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val main = hd prems 

102 
in 

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if (nprems = 1) then HOLogic.Not $ (HOLogic.dest_Trueprop main) 
15371  104 
else trans_elim (main, tl prems, concl) 
15347  105 
end; 
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(* convert an elim rule into an equivalent formula, of type term. *) 
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fun elimR2Fol elimR = 
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let val elimR' = Drule.freeze_all elimR 

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val (prems,concl) = (prems_of elimR', concl_of elimR') 

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in 

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case concl of Const("Trueprop",_) $ Free(_,Type("bool",[])) 

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=> HOLogic.mk_Trueprop (elimR2Fol_aux prems concl) 
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 Free(x,Type("prop",[])) => HOLogic.mk_Trueprop(elimR2Fol_aux prems concl) 

15347  116 
 _ => raise ELIMR2FOL("Not an elimination rule!") 
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end; 

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(* check if a rule is an elim rule *) 
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fun is_elimR th = 
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case (concl_of th) of (Const ("Trueprop", _) $ Var (idx,_)) => true 

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 Var(indx,Type("prop",[])) => true 
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 _ => false; 

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(* convert an elimrule into an equivalent theorem that does not have the 
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predicate variable. Leave other theorems unchanged.*) 

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fun transform_elim th = 
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if is_elimR th then 

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let val tm = elimR2Fol th 

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val ctm = cterm_of (sign_of_thm th) tm 

15997  132 
in 
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prove_goalw_cterm [] ctm (fn prems => [elimRule_tac th]) 
15997  134 
end 
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else th; 
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(**** Transformation of Clasets and Simpsets into FirstOrder Axioms ****) 

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(* repeated resolution *) 
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fun repeat_RS thm1 thm2 = 
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let val thm1' = thm1 RS thm2 handle THM _ => thm1 

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in 

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if eq_thm(thm1,thm1') then thm1' else (repeat_RS thm1' thm2) 

145 
end; 

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(*Convert a theorem into NNF and also skolemize it. Original version, using 
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Hilbert's epsilon in the resulting clauses.*) 

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fun skolem_axiom th = 

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let val th' = (skolemize o make_nnf o ObjectLogic.atomize_thm o Drule.freeze_all) th 
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in repeat_RS th' someI_ex 

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end; 

15347  154 

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fun cnf_rule th = make_clauses [skolem_axiom (transform_elim th)]; 
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(*Transfer a theorem into theory Reconstruction.thy if it is not already 
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inside that theory  because it's needed for Skolemization *) 
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(*This will refer to the final version of theory Reconstruction.*) 
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val recon_thy_ref = Theory.self_ref (the_context ()); 

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(*If called while Reconstruction is being created, it will transfer to the 
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current version. If called afterward, it will transfer to the final version.*) 

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fun transfer_to_Reconstruction th = 
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transfer (Theory.deref recon_thy_ref) th handle THM _ => th; 
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fun is_taut th = 
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case (prop_of th) of 
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(Const ("Trueprop", _) $ Const ("True", _)) => true 
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 _ => false; 
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(* remove tautologous clauses *) 
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val rm_redundant_cls = List.filter (not o is_taut); 
15347  176 

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(* transform an Isabelle thm into CNF *) 

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fun cnf_axiom_aux th = 
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map zero_var_indexes 
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(rm_redundant_cls (cnf_rule (transfer_to_Reconstruction th))); 
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16009  183 
(**** SKOLEMIZATION BY INFERENCE (lcp) ****) 
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(*Traverse a term, accumulating Skolem function definitions.*) 

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fun declare_skofuns s t thy = 

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let fun dec_sko (Const ("Ex",_) $ (xtp as Abs(_,T,p))) (n, (thy, axs)) = 
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(*Existential: declare a Skolem function, then insert into body and continue*) 
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let val cname = s ^ "_" ^ Int.toString n 

16012  190 
val args = term_frees xtp (*get the formal parameter list*) 
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val Ts = map type_of args 
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val cT = Ts > T 

16125  193 
val c = Const (Sign.full_name (Theory.sign_of thy) cname, cT) 
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val rhs = list_abs_free (map dest_Free args, HOLogic.choice_const T $ xtp) 
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(*Forms a lambdaabstraction over the formal parameters*) 
16009  196 
val def = equals cT $ c $ rhs 
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val thy' = Theory.add_consts_i [(cname, cT, NoSyn)] thy 

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(*Theory is augmented with the constant, then its def*) 
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val cdef = cname ^ "_def" 
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val thy'' = Theory.add_defs_i false [(cdef, def)] thy' 
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in dec_sko (subst_bound (list_comb(c,args), p)) 
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(n+1, (thy'', get_axiom thy'' cdef :: axs)) 
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end 
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 dec_sko (Const ("All",_) $ (xtp as Abs(a,T,p))) (n, thx) = 
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(*Universal quant: insert a free variable into body and continue*) 
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let val fname = variant (add_term_names (p,[])) a 
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in dec_sko (subst_bound (Free(fname,T), p)) (n, thx) end 
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 dec_sko (Const ("op &", _) $ p $ q) nthy = 
209 
dec_sko q (dec_sko p nthy) 

210 
 dec_sko (Const ("op ", _) $ p $ q) nthy = 

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dec_sko q (dec_sko p nthy) 

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 dec_sko (Const ("HOL.tag", _) $ p) nthy = 
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dec_sko p nthy 
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 dec_sko (Const ("Trueprop", _) $ p) nthy = 
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dec_sko p nthy 

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 dec_sko t nthx = nthx (*Do nothing otherwise*) 
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in #2 (dec_sko t (1, (thy,[]))) end; 
16009  218 

219 
(*cterms are used throughout for efficiency*) 

220 
val cTrueprop = Thm.cterm_of (Theory.sign_of HOL.thy) HOLogic.Trueprop; 

221 

222 
(*cterm version of mk_cTrueprop*) 

223 
fun c_mkTrueprop A = Thm.capply cTrueprop A; 

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(*Given an abstraction over n variables, replace the bound variables by free 

226 
ones. Return the body, along with the list of free variables.*) 

227 
fun c_variant_abs_multi (ct0, vars) = 

228 
let val (cv,ct) = Thm.dest_abs NONE ct0 

229 
in c_variant_abs_multi (ct, cv::vars) end 

230 
handle CTERM _ => (ct0, rev vars); 

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232 
(*Given the definition of a Skolem function, return a theorem to replace 

233 
an existential formula by a use of that function.*) 

16588  234 
fun skolem_of_def def = 
16009  235 
let val (c,rhs) = Drule.dest_equals (cprop_of (Drule.freeze_all def)) 
236 
val (ch, frees) = c_variant_abs_multi (rhs, []) 

237 
val (chil,cabs) = Thm.dest_comb ch 

16588  238 
val {sign,t, ...} = rep_cterm chil 
16009  239 
val (Const ("Hilbert_Choice.Eps", Type("fun",[_,T]))) = t 
240 
val cex = Thm.cterm_of sign (HOLogic.exists_const T) 

241 
val ex_tm = c_mkTrueprop (Thm.capply cex cabs) 

242 
and conc = c_mkTrueprop (Drule.beta_conv cabs (Drule.list_comb(c,frees))); 

243 
in prove_goalw_cterm [def] (Drule.mk_implies (ex_tm, conc)) 

244 
(fn [prem] => [ rtac (prem RS someI_ex) 1 ]) 

245 
end; 

246 

247 

248 
(*Converts an Isabelle theorem (intro, elim or simp format) into nnf.*) 

249 
fun to_nnf thy th = 

16588  250 
th > Thm.transfer thy 
251 
> transform_elim > Drule.freeze_all 

252 
> ObjectLogic.atomize_thm > make_nnf; 

16009  253 

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(*The cache prevents repeated clausification of a theorem, 

16800  255 
and also repeated declaration of Skolem functions*) (* FIXME better use Termtab!? *) 
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val clause_cache = ref (Symtab.empty : (thm * thm list) Symtab.table) 
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16009  258 
(*Declare Skolem functions for a theorem, supplied in nnf and with its name*) 
259 
fun skolem thy (name,th) = 

16588  260 
let val cname = (case name of "" => gensym "sko"  s => Sign.base_name s) 
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val (thy',axs) = declare_skofuns cname (#prop (rep_thm th)) thy 
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in (map skolem_of_def axs, thy') end; 
16009  263 

264 
(*Populate the clause cache using the supplied theorems*) 

265 
fun skolemlist [] thy = thy 

266 
 skolemlist ((name,th)::nths) thy = 

17412  267 
(case Symtab.lookup (!clause_cache) name of 
16009  268 
NONE => 
16588  269 
let val (nnfth,ok) = (to_nnf thy th, true) 
270 
handle THM _ => (asm_rl, false) 

271 
in 

272 
if ok then 

273 
let val (skoths,thy') = skolem thy (name, nnfth) 

274 
val cls = Meson.make_cnf skoths nnfth 

17412  275 
in change clause_cache (Symtab.update (name, (th, cls))); 
16588  276 
skolemlist nths thy' 
277 
end 

278 
else skolemlist nths thy 

279 
end 

16009  280 
 SOME _ => skolemlist nths thy) (*FIXME: check for duplicate names?*) 
281 

282 
(*Exported function to convert Isabelle theorems into axiom clauses*) 

15956  283 
fun cnf_axiom (name,th) = 
284 
case name of 

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"" => cnf_axiom_aux th (*no name, so can't cache*) 
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 s => case Symtab.lookup (!clause_cache) s of 
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NONE => 
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let val cls = cnf_axiom_aux th 
17412  289 
in change clause_cache (Symtab.update (s, (th, cls))); cls end 
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 SOME(th',cls) => 
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if eq_thm(th,th') then cls 
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else (*New theorem stored under the same name? Possible??*) 
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let val cls = cnf_axiom_aux th 
17412  294 
in change clause_cache (Symtab.update (s, (th, cls))); cls end; 
15347  295 

15956  296 
fun pairname th = (Thm.name_of_thm th, th); 
297 

298 
fun meta_cnf_axiom th = 

299 
map Meson.make_meta_clause (cnf_axiom (pairname th)); 

15499  300 

15347  301 

302 
(* changed: with one extra case added *) 

15956  303 
fun univ_vars_of_aux (Const ("Hilbert_Choice.Eps",_) $ Abs(_,_,body)) vars = 
304 
univ_vars_of_aux body vars 

305 
 univ_vars_of_aux (Const ("Ex",_) $ Abs(_,_,body)) vars = 

306 
univ_vars_of_aux body vars (* EX x. body *) 

15347  307 
 univ_vars_of_aux (P $ Q) vars = 
15956  308 
univ_vars_of_aux Q (univ_vars_of_aux P vars) 
15347  309 
 univ_vars_of_aux (t as Var(_,_)) vars = 
15956  310 
if (t mem vars) then vars else (t::vars) 
15347  311 
 univ_vars_of_aux _ vars = vars; 
312 

313 
fun univ_vars_of t = univ_vars_of_aux t []; 

314 

315 

316 
fun get_new_skolem epss (t as (Const ("Hilbert_Choice.Eps",_) $ Abs(_,tp,_))) = 

317 
let val all_vars = univ_vars_of t 

318 
val sk_term = ResSkolemFunction.gen_skolem all_vars tp 

319 
in 

320 
(sk_term,(t,sk_term)::epss) 

321 
end; 

322 

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(*FIXME: use alist lookup!!*) 
15531  324 
fun sk_lookup [] t = NONE 
325 
 sk_lookup ((tm,sk_tm)::tms) t = if (t = tm) then SOME (sk_tm) else (sk_lookup tms t); 

15347  326 

15390  327 
(* get the proper skolem term to replace epsilon term *) 
15347  328 
fun get_skolem epss t = 
15956  329 
case (sk_lookup epss t) of NONE => get_new_skolem epss t 
330 
 SOME sk => (sk,epss); 

15347  331 

16009  332 
fun rm_Eps_cls_aux epss (t as (Const ("Hilbert_Choice.Eps",_) $ Abs(_,_,_))) = 
333 
get_skolem epss t 

15347  334 
 rm_Eps_cls_aux epss (P $ Q) = 
16009  335 
let val (P',epss') = rm_Eps_cls_aux epss P 
336 
val (Q',epss'') = rm_Eps_cls_aux epss' Q 

337 
in (P' $ Q',epss'') end 

15347  338 
 rm_Eps_cls_aux epss t = (t,epss); 
339 

16009  340 
fun rm_Eps_cls epss th = rm_Eps_cls_aux epss (prop_of th); 
15347  341 

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(* replace the epsilon terms in a formula by skolem terms. *) 
15347  343 
fun rm_Eps _ [] = [] 
16009  344 
 rm_Eps epss (th::thms) = 
345 
let val (th',epss') = rm_Eps_cls epss th 

346 
in th' :: (rm_Eps epss' thms) end; 

15347  347 

348 

349 

15872  350 
(**** Extract and Clausify theorems from a theory's claset and simpset ****) 
15347  351 

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(*Preserve the name of "th" after the transformation "f"*) 
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fun preserve_name f th = Thm.name_thm (Thm.name_of_thm th, f th); 
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354 

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(*Tags identify the major premise or conclusion, as hints to resolution provers. 
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However, they don't appear to help in recent tests, and they complicate the code.*) 
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val tagI = thm "tagI"; 
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val tagD = thm "tagD"; 
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359 

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val tag_intro = preserve_name (fn th => th RS tagI); 
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val tag_elim = preserve_name (fn th => tagD RS th); 
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362 

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fun rules_of_claset cs = 
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let val {safeIs,safeEs,hazIs,hazEs,...} = rep_cs cs 
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val intros = safeIs @ hazIs 
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val elims = safeEs @ hazEs 
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in 
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debug ("rules_of_claset intros: " ^ Int.toString(length intros) ^ 
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" elims: " ^ Int.toString(length elims)); 
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if tagging_enabled 
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then map pairname (map tag_intro intros @ map tag_elim elims) 
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else map pairname (intros @ elims) 
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373 
end; 
15347  374 

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fun rules_of_simpset ss = 
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let val ({rules,...}, _) = rep_ss ss 
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val simps = Net.entries rules 
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in 
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debug ("rules_of_simpset: " ^ Int.toString(length simps)); 
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map (fn r => (#name r, #thm r)) simps 
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381 
end; 
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382 

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383 
fun claset_rules_of_thy thy = rules_of_claset (claset_of thy); 
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fun simpset_rules_of_thy thy = rules_of_simpset (simpset_of thy); 
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385 

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fun claset_rules_of_ctxt ctxt = rules_of_claset (local_claset_of ctxt); 
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fun simpset_rules_of_ctxt ctxt = rules_of_simpset (local_simpset_of ctxt); 
15347  388 

389 

15872  390 
(**** Translate a set of classical/simplifier rules into CNF (still as type "thm") ****) 
15347  391 

392 
(* classical rules *) 

15872  393 
fun cnf_rules [] err_list = ([],err_list) 
16009  394 
 cnf_rules ((name,th) :: thms) err_list = 
15872  395 
let val (ts,es) = cnf_rules thms err_list 
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in (cnf_axiom (name,th) :: ts,es) handle _ => (ts, (th::es)) end; 
15347  397 

398 

15872  399 
(**** Convert all theorems of a claset/simpset into clauses (ResClause.clause) ****) 
15347  400 

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fun addclause (c,th) l = 
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if ResClause.isTaut c then l else (c,th) :: l; 
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403 

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404 
(* outputs a list of (clause,thm) pairs *) 
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405 
fun clausify_axiom_pairs (thm_name,thm) = 
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let val isa_clauses = cnf_axiom (thm_name,thm) 
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val isa_clauses' = rm_Eps [] isa_clauses 
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408 
val clauses_n = length isa_clauses 
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fun make_axiom_clauses _ [] []= [] 
16897  410 
 make_axiom_clauses i (cls::clss) (cls'::clss') = 
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addclause (ResClause.make_axiom_clause cls (thm_name,i), cls') 
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(make_axiom_clauses (i+1) clss clss') 
15347  413 
in 
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414 
make_axiom_clauses 0 isa_clauses' isa_clauses 
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415 
end 
15347  416 

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fun clausify_rules_pairs [] err_list = ([],err_list) 
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418 
 clausify_rules_pairs ((name,thm)::thms) err_list = 
16897  419 
let val (ts,es) = clausify_rules_pairs thms err_list 
420 
in 

421 
((clausify_axiom_pairs (name,thm))::ts, es) 

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handle THM (msg,_,_) => 
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(debug ("Cannot clausify " ^ name ^ ": " ^ msg); 
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(ts, (thm::es))) 
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425 
 ResClause.CLAUSE (msg,t) => 
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426 
(debug ("Cannot clausify " ^ name ^ ": " ^ msg ^ 
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": " ^ TermLib.string_of_term t); 
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428 
(ts, (thm::es))) 
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429 

16897  430 
end; 
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431 

15347  432 

16009  433 
(*Setup function: takes a theory and installs ALL simprules and claset rules 
434 
into the clause cache*) 

435 
fun clause_cache_setup thy = 

436 
let val simps = simpset_rules_of_thy thy 

437 
and clas = claset_rules_of_thy thy 

438 
in skolemlist clas (skolemlist simps thy) end; 

439 

16563  440 
val clause_setup = [clause_cache_setup]; 
441 

442 

443 
(*** meson proof methods ***) 

444 

445 
fun cnf_rules_of_ths ths = List.concat (#1 (cnf_rules (map pairname ths) [])); 

446 

447 
fun meson_meth ths ctxt = 

448 
Method.SIMPLE_METHOD' HEADGOAL 

449 
(CHANGED_PROP o Meson.meson_claset_tac (cnf_rules_of_ths ths) (local_claset_of ctxt)); 

450 

451 
val meson_method_setup = 

452 
[Method.add_methods 

453 
[("meson", Method.thms_ctxt_args meson_meth, 

454 
"The MESON resolution proof procedure")]]; 

15347  455 

456 
end; 